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Complexity Theory Complexity Theory IE 661: Scheduling Theory Fall 2003 Satyaki Ghosh Dastidar Outline z Goals z Computation of Problems { Concepts and Definitions z Complexity { Classes and Problems z Polynomial Time Reductions { Examples and Proofs z Summary University at Buffalo Department of Industrial Engineering 2 Goals of Complexity Theory z To provide a method of quantifying problem difficulty in an absolute sense. z To provide a method comparing the relative difficulty of two different problems. z To be able to rigorously define the meaning of efficient algorithm. (e.g. Time complexity analysis of an algorithm). University at Buffalo Department of Industrial Engineering 3 Computation of Problems Concepts and Definitions Problems and Instances A problem or model is an infinite family of instances whose objective function and constraints have a specific structure. An instance is obtained by specifying values for the various problem parameters. Measurement of Difficulty Instance z Running time (Measure the total number of elementary operations). Problem z Best case (No guarantee about the difficulty of a given instance). z Average case (Specifies a probability distribution on the instances). z Worst case (Addresses these problems and is usually easier to analyze). University at Buffalo Department of Industrial Engineering 5 Time Complexity Θ-notation (asymptotic tight bound) fn( ) : there exist positive constants cc12, , and n 0 such that Θ=(())gn 0≤≤≤cg12 ( n ) f ( n ) cg ( n ) for all n ≥ n 0 O-notation (asymptotic upper bound) fn( ) : there exist positive constants c and n0 such that Ogn(())= 0≤≤fn ( ) cgn ( ) for all n ≥ n0 Ω-notation (asymptotic lower bound) fn( ) : there exist positive constants c and n0 such that Ω=(())gn 0≤≤cg ( n ) f ( n ) for all n ≥ n0 o-notation (asymptotic “loose” upper bound) fn( ) : for any positive constant c> 0, there exists a constant ogn(())= nfncgnnn00>≤<≥0 such that 0 ( ) ( ) for all University at Buffalo Department of Industrial Engineering 6 …Time Complexity (contd.) ω-notation (asymptotic “loose” lower bound) fn( ) : for any positive constant c> 0, there exists a constant ω(())gn = ncgnfnnn00>≤<≥0 such that 0 ( ) ( ) for all cgn2 () cg()n f ()n f ()n f ()n cg() n cg1 () n n n n n0 n0 n0 f ()n =Θ (())g n f ()nO= (())g n f ()n = Ω (())g n University at Buffalo Department of Industrial Engineering 7 Algorithm Types z Polynomial Time Algorithm: An algorithm whose running time is bounded by a polynomial function is called a polynomial time algorithm. Example: Shortest path problem with nonnegative weights. Running Time: O(n2) z Exponential Time Algorithm: An algorithm that is bounded by an exponential function is called an exponential time algorithm. Example: Check every number of n digits to find a solution. Running Time: O(10n) z Pseudopolynomial Time Algorithm: A pseudopolynomial time algorithm is one that is polynomial in the length of the data when encoded in unary. Example: Integer Knapsack Problem. Running time: O(nb) University at Buffalo Department of Industrial Engineering 8 Turing Machine z A Turing machine is an abstract representation of a computing device. The behavior of a TM is completely determined by: • The state the machine is in, • The number on the square it is scanning, and • A table of instructions or the transition table. “A function is computable if it can be computed by a Turing Machine.” - Church-Turing Hypothesis University at Buffalo Department of Industrial Engineering 9 Finite State Machine State Read Write Move Next State 0 0 L S1 S1 B 1 L S2 1 B R S1 0 1 R S2 S2 B 0 R S2 1 1 L S1 State Transition Table for a Turing Transition State Diagram for Machine Turing Machine University at Buffalo Department of Industrial Engineering 10 Decision Problem Decision problems are those that have a TRUE/FALSE answer. z SATISFIABILITY: Given a set of variables and a collection of clauses defined over the variables, is there an assignment of values to the variables for which each of the clauses is true? Example: Consider the expression ()()()()xxxxxxxxxxxxxxxx1432124323155142+++ +++ +++ +++ It can be easily verified that the assignment x1=0, x2=0, x3=0, x4=0, and x5=0 gives a truth assignment to each one of the four clauses. University at Buffalo Department of Industrial Engineering 11 Decision Problems and Reductions z For every optimization problem there is a corresponding decision problem. Example: Fm||Cmax minimize makespan (optmization). Is there a schedule with a makespan ≤ z ? (decision). Problem Reduction: Problem P reduces to problem P′ if for any instance of P an equivalent instance of P′ can be constructed. Polynomial Reducibility: Problem P polynomially reduces to problem P′ if a polynomial time algorithm for P′ implies polynomial time algorithm for P. P∝P′ University at Buffalo Department of Industrial Engineering 12 Complexity Classes and Problems Complexity Classes z Definition: (Class P) The class P contains all decision problems for which there exists a Turing machine algorithm that leads to the right “yes/no” answer in a number of steps bounded by a polynomial in the length of the encoding. z Definition: (Class NP) The class NP contains all decision problems for which, given a proper guess, there exists a polynomial time “proof” or “certificate” C that can verify if the guess is the right “yes/no” answer. NP P A tentative view of the world of NP University at Buffalo Department of Industrial Engineering 14 … Complexity Classes (contd.) z Definition: (Class co-P) The class co-P contains all decision problems for which there exists a polynomial time algorithm that can determine what all “yes/no” answers are incorrect. z Definition: (Class co-NP) The class co-NP contains all decision problems such that there exists a polynomial time “proof” or “certificate” C that can verify if the problem does not have the right “yes/no” answer. co-NP NP P A view of the world of NP and co-NP University at Buffalo Department of Industrial Engineering 15 Important Results z P = co-P z NP ≠ co-NP z P ≠ NP It turns out that almost all interesting problems lie in NP and P is the set of easy problems. So are all interesting problems easy, i.e. do we have P = NP? This is the main open question in Computer Science. It is like other great questions z Is there intelligent life in the universe? z What is the meaning of life? z Will you get a job when you graduate? University at Buffalo Department of Industrial Engineering 16 NP-Complete Problems z Definition: (NP-complete) A decision problem D is said to be NP-complete if DNP and, for all other decision problems D′NP, there exists a polynomial transformation from D′ to D, i.e., D′∝ D. Assumption: P≠NP. Result: If any single NP-complete problem can be solved in polynomial time, then all problems in NP can be solved. Cook’s Theorem NP co-“NP-complete” A problem is NP-complete if: P (i) The problem is in NP “NP-complete” co-NP (ii) All other problems in NP polynomially transforms into The world of NP, revisited the above problem. University at Buffalo Department of Industrial Engineering 17 NP-Hard Problems z Definition: (NP-hard) A decision problem whether a member of NP or not, to which we can transform a NP-complete problem is at least as hard as the NP-complete problem. Such a decision problem is called NP-hard. Example: KTH LARGEST SUBSET A∈{,aa ,… a } : Given a set 12 t ,ba ≤ ∑ j , jA∈ andk ≤ 2,||A do there exist at least K distinct subsets where ′ A ∈{SS12,,… SK } and A′ ⊆ A such that ∑ Sbj ≤ ? jA∈ ′ University at Buffalo Department of Industrial Engineering 18 Six Basic NP-Complete Problems z 3-SATISFIABILITY: Given a collection C = {c1, c2, …, cm} of clauses on a finite set U of variables such that |ci|=3 for 1≤i≤m, is there a truth assignment for U that satisfies all the clauses in C? z 3-DIMENSIONAL MATCHING: Given a set M ⊆×× WXY , where W, X, and Y are disjoint sets having the same number q of elements, does M contain a matching, i.e., a subset / / / z M ⊆ M such that | M | = q and no two elements of M agree in any coordinate? 1 t a z PARTITION: Given positive integers a1, … , at and b = ∑ j , 2 j=1 do there exist two disjoint subsets S1 and S2 such that ∑ abj = for i= 1, 2 ? jS∈ i University at Buffalo Department of Industrial Engineering 19 …Six Basic Problems (contd.) z VERTEX COVER: Given a graph G=(V,E) and a positive integer K ≤ |V|, is there a vertex cover of size K or less for G, i.e., a subset VV/ ⊆ such that | V /| ≤ K and, for each edge {,}uv∈ E ,at least one of u and v belongs to V /? z HAMILTONIAN CIRCUIT: For a graph G = (N, A) with node set N and arc set A, does there exist a circuit (or tour) that connects all the N nodes exactly once? z CLIQUE: For a graph G = (N, A) with node set N and arc set A, does there exist a clique of size c? i.e., does there exist a setNN* ⊂ , consisting of c nodes such that for each distinct pair of nodes uv, ∈ N,* the arc {u,v} is an element of A? University at Buffalo Department of Industrial Engineering 20 Transformation Topology SATISFIABILITY 3-SAT 3DM VC PARTITION HC CLIQUE Diagram of the sequence of transformations used to prove that the six basic problems are NP-complete. Problems of which the complexity is established through a reduction from PARTITION typically have pseudopolynomial time algorithms and are therefore NP-hard in the ordinary sense.
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